Find all natural numbers $ x,y$ such that $ y| (x^{2}+1)$ and $ x^{2}| (y^{3}+1)$.

Suppose $x, y, a$ are real numbers such that $x+y = x^3 +y^3 = x^5 +y^5 = a$. Find all possible values of $a.$

In an acute-angled triangle $ABC$, $M$ and $N$ are points on the sides $AC$ and $BC$ respectively, and $K$ the midpoint of $MN$. The circumcircles of triangles $ACN$ and $BCM$ meet again at a point $D$. Prove that the line $CD$ contains the circumcenter $O$ of $\triangle ABC$ if and only if $K$ is on the perpendicular bisector of $AB.$

If $a, b, c$ are non-negative real numbers with $ a^2 + b^2 + c^2 = 1$, prove that: \[ \frac {a}{b^2 + 1} + \frac {b}{c^2 + 1} + \frac {c}{a^2 + 1} \geq \frac {3}{4}(a\sqrt {a} + b\sqrt {b} + c\sqrt {c})^2\]